What are the model assumptions for a poisson mixed effects model? tldr; Aside from equidispersion, what are the model assumptions I should be checking for in a poisson mixed effects model that has a random intercept, group mean centered transformations of its explanatory variables, and group means included as variables?
I have a county-year dataset with 21 counties that each have 8 years of data (N = 168). I am attempting to model the number of prescription opioid related hospitalizations for this data using the number of prescription opioid pills supplied to each county in a year, the prescription rate of each county in a year, and demographic and economic (unemployment rate, median household income) variables also at the county-year level.
I am using a poisson distribution (with a log link) and the lme4 package to estimate this model with a random intercept, group mean centered transformations of each variable, and the group mean of each variable in line with the specification found in Bell and Jones, 2015 . Additionally, I have been using the DHARMa package to visually examine the relationships between my explanatory variables and the randomized quantile residuals from the model and to test for overdispersion.
I am not formally trained in mixed effects modeling and want to be as sure as one can be that the estimates from my model are not biased. What are the assumptions of a poisson mixed effects model and is there a rigorous set of steps for testing these assumptions (either by looking at residuals or any other part of the model output)?
Thank you in advance for any help!
 A: If you're using R, this is a great place to start for count models: https://cran.r-project.org/web/packages/pscl/vignettes/countreg.pdf
You'll be able to see all the models/packages available and their accompanying assumptions.
For Poisson, the main model assumptions are:

*

*The variance is equal to the mean. As you have stated, overdispersion can occur when your variance is much higher than your mean. DHARMa provides effective tools to quantify overdispersion, and when it is present, you would want to consider quasi-Poisson, negative binomial, or zero-inflated models (see CRAN package link above).


*The response variable is non-negative integer data.


*The responses are independent from one another.


*The responses occur over fixed time or space.
When you are incorporating mixed effects, you want to make sure that your random effect is some kind of grouping variable (perhaps a county ID in your case). You specify this to account for any in-county differences that may account for the number of opioid-related hospitalizations.
An overview of linear mixed models, and how to specify random effects can be found here: https://www.youtube.com/watch?v=QCqF-2E86r0&t=95s
Hope this helps.
